Summation

[Rikeus]

Sorry for the vague subject title, but I couldn't think of anything halfway between 'so vague there may as well be no title' and 'entire post is in the subject line'
Also, I couldn't find anything online to help me format this in LaTeX, so I'll just have to use plaintext.

Anyway, I learned about the summation function in school recently, and it got me thinking. I wondered if an expression of the form
x=n
Σ f(x)
x=c
Where c is a constant and n and x are variables could be rewritten as a function of n.
To test this, I asked wolfram alpha to sum from x=1 to x=n 2x, and it returned x^2 + x. I tried this with several other functions, including sin(x) and e^x, and it worked for them too. I did notice that there were some functions that didn't work with it, such as tan(x) and x^x. I also noticed that whenever f(x) was polynomial, f(n) was a polynomial with 1 extra power (eg, linear became quadratic, quadratic became cubic, x^20 became a very big function involving x^21).

What I was wondering was, how is this done? The only way I could think of to do this would be by some sort of regression using a bunch of different values for n, or in the case of the polynomials just taking however many points are required to define the curve.

[gorcee]

I think it's a little confusing how you're using x for both the summation index and the independent variable. Is this what you mean? If so, you need to be careful to recognize that your results really only make sense for integer values of x.

Take your example \sum_{x=1}^{n}2x. What does this equal? Well, we can pull out the 2 due to the distributive property and get \sum_{x=1}^{n}2x = 2 \sum_{x=1}^{n}x. Let's look at the sum on the right hand side. What is this? This is just 1 + 2 + 3 + 4 + ... + n, or the sum of all integers from 1 to n. There's a very clean formula for that: 1+2+3+...+n-1+n = n(n+1)/2. But don't forget that we multiply by 2, so we get n(n+1) = n^2+n. This is not the same as x^2+x, however.

In general, can you take a function that is the sum of something and make it a function of the upper limit of the sum? Sure, but keep in mind that the function you come up with is probably not going to be well-defined, and even if it is, you have to be careful to consider it's domain.

[gmalivuk]

Where c is a constant and n and x are variables

In that summation x is just a dummy variable, and shouldn't come out in your answer at all. For example,

\sum_{x=1}^{n}x=\frac{n(n+1)}{2}

[Coding]

Now that gorcee and gmalivuk have posted the formula for the summation, you can try proving that it holds for any natural number n. If you are interested in that sort of thing, anyway.

[Qaanol]

I also noticed that whenever f(x) was polynomial, f(n) was a polynomial with 1 extra power (eg, linear became quadratic, quadratic became cubic, x^20 became a very big function involving x^21).

Correct.

[Ben-oni]

I prefer the methods of finite calculus for these problems. Easier to remember (once you know it?), and allows "summation by parts", much like integration by parts.

Also, you may have stumbled upon something fairly important in ∑ⁿ_x=1sin(x)... remember that one!

[moiraemachy]

To answer the OP's question: mathematical induction is what you're looking for. I won't say anything else because it may spoil the fun. If you have any questions ask.